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Time: 2 1 / 2 hours The University of British Columbia
Final Examination April 2004
Math 152 Instructors: J. Miller (201), D. Burggraf (202), L. Scull (203), R. Fmese (204), D. Coombs (205), J. Fournier
(206), J. Solymosi (207), M. Pakzad (208) Special Instructions: No books, calculators or formula sheets allowed. Write all your answers in the answer booklet(s). Write your name and student number and the number of booklets used on each booklet.
Show enough work to justify your answers. 1. [20 points] Multiple choice. Write your answer in your answer booklet like this: a.Xb.Xc.Xd.Xe.Xf.Xg.IXh.X
where X is your choice of A,B,C or D. (a) The following augmented matrix is given for a system of equations. 1 0 6 3
0 1 —2 —6
0 0 0 0
If the system in consistent, ﬁnd the general solution. Otherwise state that there is no solution. 931 = 3 '— 5$3 $1 2 3  5213 $1 = 3 ' 633
A) No solution B) 9:2 = —6 + 233 C) 9:2 = —6 + 2333 D) 2:2 : a free parameter
9:3 = 0 m3 = a free parameter :33 = 3 + (1/2)a:2 1 0 0
(b) Let e1 = [0] , e2 2 [1:1 and e3 = Suppose that T is a linear transformation from R3 into R2
U such that Tel) = [*4] Tea) = [3] Tea) = [‘3‘] 181
Find a formula for T(x) for an arbitrary x = [$31 in R3.
933 :31 31131 + 3:132 " 3933 $1 3331 — 4:32
A) T $2 = 3561 B) T .172 3271
m3 —4x1 + 503 3:3 332 + 9:3 0) 2 [saws—3:133] D) =[3ml—4xg] —4s: + a: 331
$3 1 3 $3 [I O (c) Find the matrix product AB, if it is deﬁned, where [ii ‘3] [1? 3]
A) [it 31 B) [—3 ‘3] C) [3 2%] D) [33 21
(d) Find the inverse of the matrix A, if it exists. 4—25
A=4m13 8—38 4 4 8 1 0 1/4
A) 24—1: —2 *1 —3 B) A“1=[0 1 4] 5 3 8 0 3/4 0
1 0 1/4
C) Ail: 1 0 —2 D) Ail doesnot exist
0 0 0 (e) Find the determinant of the matrix 1 —3 5
4 —4 —2
5 3 2 A} —48 B) 152 C) 212 D) —212 (f) Calculate the area of the parallelogram with vertices (0, 0), (4, 8), (10, 10) and (6, 2). A) 39 B) 40 C) 44 D) so (g) Let A be a 5 X 5 matrix with characteristic polynomial
det(A — AI) = 415 + 4x‘ + 45213.
Find the eigenvalues and their (algebraic) multiplicities. A) 0 with multiplicity 1 — 9 with multiplicity 1 5 with multiplicity 1
B) 0 with multiplicity 3  9 with multiplicity 1 5 with multiplicity 1
C) 0 with multiplicity 1 — 5 with multiplicity 1 9 with multiplicity 1
D) 0 with multiplicity 3 — 5 with multiplicity 1. 9 with multiplicity 1 v} (h) Diagonalize the matrix A, if possible. That is, ﬁnd an invertible matrix P and a diagonal matrix D such that A = PDP"1. _
1 1 4 A = 0 —4 0 —5 —1 «8 10—1 —40—3 10e1 —400
A) P: —9 ~41 0,1): 0 —4 0 B) P: m9 —4 0,1): 0 —4 0
1 11 0—4—3 111 00—3
10—1 —400 1—9—1 410
C)P= 0—4 0,1): 1} 1 0 D) P: ~9 —4 0,1): 0—4 0
111 00—3 1—41 sews 2. [16 points] Let P be the plane given by the equation a: + 2p + z = 3 and Q be the plane given by the
equation a: + y + z = 2. Let q = (l, —1,0). (a) Do the planes P and Q intersect? If so ﬁnd an equation in parametric form for their line of intersection.
(b) Find an equation for the plane that does not intersect P and passes through q. (c) Find the point on Q that is closest to q. 3. [16 points] Let R be the linear transformation acting on vectors in two dimensions that reﬂects a vector across the line as = y, and let S be the linear transformation, also acting on vectors in two dimensions, that '
reﬂects vectors across the y axis. (a) Find the matrices for R and S
(b) Find the matrix of the linear transformation deﬁned by ﬁrst performing the reﬂection R and then the
reﬂection S. Identify this transformation as either another reﬂection, a rotation or a projection. (c) Is it possible that the linear transformation obtained by ﬁrst rotating :1. vector and then reﬂecting it could
be a projection? Give a reason. 4. [16 points] Consider the system of linear equations :1: ~23; +32. = 2
9: +2y + z 2 1
2y +32 = b Determine the values of a and b for which this system has (a) a unique solution, (b) no solution and (c)
inﬁnitely many solutions. 5. [16 points] Let A, B and C be three cities whose populations in year as are given by by 3(a), b(n) and
C(n), n = 0,1,2. . .. Every year people move from city to city according to the following rules: 1/ 2 of the residents of A stay in A, 1/4 of the residents of A move to B, 1/4 of the residents of A move to C
1/ 4 of the residents of B move to A, 1/2 of the residents of B stay in B, 1/4 of the residents of B move to C’
1/ 4 of the residents of 0 move to A, 1 /4 of the residents of C move to B, 1/ 2 of the residents of 0 stay in C If the initial populations are (1(0) = 10,000, 5(0) _= 20, 000, 0(0) 2 30,000
(a) Find the populations a(2), M2} and c(2) after 2 years
(b) Find the limiting populations a(n), b(n) and c(n) when n becomes large. 6. [16 points] Suppose two variables I(t) and V(t) in a circuit satisfy the system of differential equations I’(t) = —I(t) — V(t)
V’(t) : I(t) — Va).
If their initial values are given by I (0) = 2 and V(0) = —3 ﬁnd the values of I(t) and V6) for all values of t. ...
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